In U.S. Pat. No. 5,060,542, there is described an apparatus and method for making and breaking drill pipe joints, and which is a major improvement over prior art. However, the torques generated during making and breaking of the joints are enormous. It would be highly desirable and important to achieve jaws that are more strong, more precision, more rugged, more symmetrical, more easily adjusted, more stable, etc., than are the jaws described in the cited patent.